Sorry for the delay, (time difference for me here in the UK).
That's strange. I don't use Excel 2007 myself as I am still on Excel 2002, (XP), but the formula should still work because it is just a simple IF statement. Without your actual figures, I can't check your spreadsheet's math, so I did some more testing myself.
Since your target cells are spread out and are not consecutive, I tested a couple of ways. I used the same cells as you have quoted, but with my own figures, eg;
E3= 10, I3= -50, M3= 23, Q3= 0, U3= -420, Y3= -510, AC3= 740, AG3= 5, AK3= 44, AO3= -2, AS3= 410
Then I copied these to B12 to B22 to create a simple, one column, list, eg;
B12= E3
B13= I3
B14= M3
B15= Q3
B16= U3
B17= Y3
B18= AC3
B19= AG3
B20= AK3
B21= AO3
B22= AS3
Doing a SUM function on (B12:B22) gives 250, so that is my test figure. It adds them all, including zeros and minus figures.
Using the formula I gave on the Row 3 figures, eg;
=0+IF(E3>0,E3,0)+IF(I3>0,I3,0)+IF(M3>0,M3,0)+IF(Q3>0,Q3,0)+IF(U3>0,U3,0)+IF(Y3>0,Y3,0)+IF(AC3>0,AC3,0)
+IF(AG3>0,AG3,0)+IF(AK3>0,AK3,0)+IF(AO3>0,AO3,0)+IF(AS3>0,AS3,0)
That gives me 1232. That means it just adds up the positive numbers.
As a double check, using a similar formula on the Col B figures, eg;
=0+IF(B12>0,B12,0)+IF(B13>0,B13,0)+IF(B14>0,B14,0)+IF(B15>0,B15,0)+IF(B16>0,B16,0)+IF(B17>0,B17,0)
+IF(B18>0,B18,0)+IF(B19>0,B19,0)+IF(B20>0,B20,0)+IF(B21>0,B21,0)+IF(B22>0,B22,0)
That gives me 1232, the same result.
So, I am mystified. Can I ask, where do your figures in the Row 3 cells derive from? If there is anything unusual about those cell contents, that might account for the discrepancy. But I can't see what would cause that.
Can you do an error check audit? Excel 2007 has a similar function to Excel 2002 called "Formula Auditing > Evaluate Formula", which can step you through what Excel evaluates each part of the formula to.
That Formula Auditing isn't in the same place in E2007 that it is in E2002, so have a look at the guidance in the link below, to get you there;
http://www.dummies.com/how-to/content/understanding-excel-2007s-formula-auditing-tools.html
It's called "Dummies.com". Don't worry about that name though, I'm sure it's not meant to be personal, 
Can you step through your formula, and see if you can find out where E2007 is making any errors? In truth, Excel doesn't normally make errors and we generally find that there's an error or unexpected consequence of a cell we completed.
Let me know if you find anything.
Mark